While this page is focused on the particular history and role of House Decados' Order of the Mantis, much of what is said here is also applicable to Questing Knights, and some of it is applicable to Swords of Lextius or other orders of Church Knights.
History and Purpose
Mantis Knights - those (largely Decados) members of the Order of the Mantis, a chivalric order sponsored by Prince Hyram - are noblemen and women who are members of an adventuring order of knights errant. Adventuring orders broadly trace their origins largely back to the order of pilgrim knights that later became the Brother Battle, and have seen their popularity rise and fall over the last thousand years. Presently, chivalric orders are very much in vogue due to the prevalence and popularity of the Emperor's Questing Knights.
The Order of the Mantis was founded by House Decados and has been positioned as a rival order to the Company of the Phoenix - indeed, the position of Prince Hyram is that the Mantis Knights are 'just as good as' the Questing Knights. Most (though not all) Mantis Knights are Decados, and while they engage in the same general sort of activity as the Questing Knights - investigating old ruins, intervening in local disputes, et cetera - they do so largely with the interests of House Decados (or at least its glorification) in mind.
Make no mistake, however - while the order serves a political purpose, the Mantis Knights are not a breeding-ground for would-be politicos. Mantis Knights /are/ adventuring knights, and like all adventuring orders they are principally concerned with investigating mysteries, exploring untamed wildernesses and ruins and retrieving artifacts from the clutches of villains. Indeed, protecting innocents is a common pastime for Mantis Knights, provided there's a spotlight on the issue - every wrong a Mantis Knight rights is fuel for the Decados public relations machine, and so more than other members of House Decados the Mantis Knights tend to be the more moral members of the House - though they are expected to be the most loyal, and when a choice arises between doing what's right and doing what's right for House Decados they are expected to choose Mantis above morals.
Unlike many other adventuring orders, Mantis Knights also often have a military role. The original Mantis Knights were chosen from House Decados' most successful commanders, and to this day many Mantis Knights serve as elite commanders in the military, where their martial victories can be held up as successes for the House. Military prowess is something that earns a great deal of respect among the nobility, and so making successful commanders Mantis Knights (and vice versa) allows the House to use the free publicity membership in an adventuring order provides to highlight their success on the battlefield. Indeed, in some cases - like the Mantis Guard on Pandemonium - Mantis Knights make up the officer corps of entire units, though even then they are also expected to continue their adventuring ways.
Mantis Knights also have the ability to requisition soldiers from local Decados vassals when required for some quest or event they are undertaking. By the Prince's command, these requests are not to be refused - but this is a command to nobles, not soldiers. A particularly charismatic Mantis Knight may be able to walk into a barracks and stir the soldiers within to action, but by the letter of their charter they are to go to the lord who mustered these soldiers and ask him for troops, at which point he will assign them to the Mantis Knight. Mantis Knights do not break the chain of command - they just have power over the lords who sit at that chain's top. Of course, if their requests are outrageous the mustering lord may defy the Prince and not provide the soldiers; this may have dire consequences for that lord, however, if Prince Hyram decides the Mantis Knight was justified after all.
Mantis Knights and the Political System
In general, Mantis Knights - and members of other adventuring orders - operate 'outside the system' rather than within it. Adventuring orders are intended to empower a class of knights who would otherwise be third wheels in the feudal system as direct actors reporting to a Prince, Emperor, Patriarch or the like, and as a consequence these orders are full of younger sons or nobles with lost fiefs. It is very rare for members of an adventuring order to hold lands or rank above knight, as that ties them to local liege lords more than to the order's patron. Politically, members of adventuring orders are nuisances - they have no ties to local political authorities and yet have the power to interfere in local affairs, giving them a boost in the beginning but ultimately preventing them from becoming an established part of a political system. This is exactly what it is intended; when Prince Hyram founded the Order of the Mantis, he intended to create a group of knights loyal to him who could not easily be swayed by local concerns or the promise of local power, and that could glorify his name and the name of House Decados with limited interference from the lords on the ground.